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v1 Financial Maths, STD2 F4 2022 HSC 27

A business buys a delivery van for $60 000. The two depreciation methods under consideration are the declining-balance method and the straight-line method.

  1. For the declining-balance method, the salvage value of the van after `n` years is given by the formula
  2. `S=V_(0)xx(0.75)^(n),`
  3. where `S` is the salvage value and `V_(0)` is the initial cost of the asset.
    1. What is the annual rate of depreciation used in this formula?  (1 mark)

    2. Calculate the salvage value of the van after 2 years, based on the given formula.  (1 mark)

  4. Using the straight-line method, the van depreciates by 13.75% of its purchase price each year.
  5. After how many full years will the value of the van be equal to or less than the value found in part (a) (ii)?  (2 marks)

Show Answers Only
  1.  i. `25text{%}`
  2. ii. `$33\ 750`
  3. `3\ text{years}`
Show Worked Solution

a.i.  `text{Depreciation rate}\ = 1-0.75 = 0.25 = 25text{%}`

a.ii.  `text{Find}\ \ S\ \ text{when}\ \ n=2:`

`S` `= V_0 xx (0.75)^2`
  `= 60\ 000 xx 0.5625`
  `= $33\ 750`

b.  `text{Using the SL method}`

`S_n` `= 60\ 000-(0.1375 xx 60\ 000) xx n`
  `= 60\ 000-8250n`

`text{Set}\ \ S_n \leq 33\ 750`

`33\ 750` `= 60\ 000-8250n`
`8250n` `= 26\ 250`
`n` `= 26\ 250 / 8250 = 3.18`
  `⇒ 3\ text{years (to nearest full year)}`
♦♦ Solid performance expected if depreciation concepts are understood.
HINT: Watch out for rounding when comparing values in part (b).

v1 Financial Maths, STD2 F4 2005 HSC 26a

A new high-end coffee machine is purchased for $25 000 in January 2020.

At the end of each year, starting in 2021, the machine depreciates in value by 15% per annum, using the declining balance method of depreciation.

In which year will the value of the machine first fall below $15 000? (2 marks)

Show Answers Only

`text(The value falls below $15 000 in the fourth year)`

`text{which will be during 2024.}`

Show Worked Solution

`text(Using the formula:)\ \ S = V_0(1-r)^n`

`text(where) \ \ V_0 = 25\ 000,\ \ r = 0.15`

`text(If)\ n = 1\ \text{(2021)}`

`S` `= 25\ 000(0.85)^1`
  `=21\ 250`

 

`text(If)\ n = 2\ \text{(2022)}`

`S` `=25\ 000(0.85)^2`
  `=25\ 000(0.7225)`
  `=18\ 062.50`

 

`text(If)\ n = 3\ \text{(2023)}`

`S` `=25\ 000(0.85)^3`
  `=25\ 000(0.614125)`
  `=15\ 353.13`

 

`text(If)\ n = 4\ \text{(2024)}`

`S` `=25\ 000(0.85)^4`
  `=25\ 000(0.52200625)`
  `=13\ 050.16`

 

`:.\ \text{The value first falls below $15 000 in the fourth year}`

`text{which will be during 2024.}`

Financial Maths, STD2 F4 2022 HSC 27

A company purchases a machine for $50 000. The two methods of depreciation being considered are the declining-balance method and the straight-line method.

  1. For the declining-balance method, the salvage value of the machine after `n` years is given by the formula
  2.     `S=V_(0)xx(0.80)^(n),`
  3. where `S` is the salvage value and `V_(0)` is the initial value of the asset.
  4.  i. What is the annual rate of depreciation used in this formula?  (1 mark)
  5. ii. Calculate the salvage value of the machine after 3 years, based on the given formula.  (1 mark)
  6. For the straight-line method, the value of the machine is depreciated at a rate of 12.2% of the purchase price each year.
  7. When will the value of the machine, using this method, be equal to the salvage value found in part (a) (ii)?  (2 marks)
Show Answers Only
  1.  i. `20text{%}`
  2. ii. `$25\ 600`
  3. `text{4 years}`
Show Worked Solution

a.i.  `text{Depreciation rate}\ = 1-0.8=0.2=20text{%}`
 

a.ii.  `text{Find}\ \ S\ \ text{when}\ \ n=3:`

`S` `=V_0 xx (0.80)^n`  
  `=50\ 000 xx (0.80)^3`  
  `=$25\ 600`  

 
b.   `text{Using the SL method}`

`S_n` `= 50\ 000-(0.122 xx 50\ 000)xxn`  
  `=50\ 000-6100n`  

 

`text{Find}\ \ n\ \ text{when}\ \ S_n=$25\ 600`

`25\ 600` `=50\ 000-6100n`  
`6100n` `=24\ 400`  
`n` `=(24\ 400)/6100`  
  `=4\ text{years}`  

♦♦ Mean mark (a.i.) 24%.
COMMENT: A poor State result in part (a.i.) that warrants attention.
 
♦ Mean mark part (b) 38%.

Financial Maths, STD2 F4 2005 HSC 26a

A sports car worth $150 000 is bought in December 2005.

In December each year, beginning in 2006, the value of the sports car is depreciated by 10% using the declining balance method of depreciation.

In which year will the depreciated value first fall below $120 000?   (2 marks)

Show Answers Only

`text(The value falls below $120 000 in the third year)`

`text{which will be during 2008.}`

Show Worked Solution

`text(Using)\ \ S = V_0(1-r)^n`

`text(where)\ \ V_0 = 150\ 000, r = text(10%)`

`text(If)\ \ n = 2,`

`S` `= 150\ 000(1-0.1)^2`
  `= 121\ 500`

 
`text(If)\ \ n= 3,`

`S` `= 150\ 000(1-0.1)^3`
  `= 109\ 350`

 

`:.\ text(The value falls below $120 000 in the third year)`

`text{which will be during 2008.}`